The properties of a new fractional g-Laplacian Monge-Ampère operator and its applications

Abstract

Abstract In this article, we first introduce a new fractional g g -Laplacian Monge-Ampère operator: F g s v ( x ) ≔ inf P.V. ∫ R n g v ( z ) − v ( x ) ∣ C − 1 ( z − x ) ∣ s d z ∣ C − 1 ( z − x ) ∣ n + s ∣ C ∈ C , {F}_{g}^{s}v\left(x):= \inf \left\{\hspace{0.1em}\text{P.V.}\hspace{0.1em}\mathop{\int }\limits_{{{\mathbb{R}}}^{n}}g\left(\frac{v\left(z)-v\left(x)}{{| {C}^{-1}\left(z-x)| }^{s}}\right)\frac{{\rm{d}}z}{{| {C}^{-1}\left(z-x)| }^{n+s}}| C\in {\mathcal{C}}\right\}, where g g is the derivative of a Young function and the diagonal matrix C {\mathcal{C}} is positive definite, which has a determinant equal to 1. First, we establish some crucial maximum principles for equations involving the fractional g g -Laplacian Monge-Ampère operator. Based on the maximum principles, the direct method of moving planes is applied to study the equation involving the fractional g g -Laplacian Monge-Ampère operator. As a result, the nonexistence of the positive solutions, symmetry, monotonicity, and asymptotic property of solutions are obtained in bounded/unbounded domains.